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Perturbing Rational Harmonic Functions by Poles

Perturbing Rational Harmonic Functions by Poles We study how adding certain poles to rational harmonic functions of the form $$R(z)-\overline{z}$$ R ( z ) - z ¯ , with $$R(z)$$ R ( z ) rational and of degree $$d\ge 2$$ d ≥ 2 , affects the number of zeros of the resulting functions. Our results are motivated by and generalize a construction of Rhie derived in the context of gravitational microlensing ( arXiv:astro-ph/0305166 ). Of particular interest is the construction and the behavior of rational functions $$R(z)$$ R ( z ) that are extremal in the sense that $$R(z)-\overline{z}$$ R ( z ) - z ¯ has the maximal possible number of $$5(d-1)$$ 5 ( d - 1 ) zeros. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

Perturbing Rational Harmonic Functions by Poles

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Publisher
Springer Journals
Copyright
Copyright © 2014 by Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Analysis; Computational Mathematics and Numerical Analysis; Functions of a Complex Variable
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/s40315-014-0083-x
Publisher site
See Article on Publisher Site

Abstract

We study how adding certain poles to rational harmonic functions of the form $$R(z)-\overline{z}$$ R ( z ) - z ¯ , with $$R(z)$$ R ( z ) rational and of degree $$d\ge 2$$ d ≥ 2 , affects the number of zeros of the resulting functions. Our results are motivated by and generalize a construction of Rhie derived in the context of gravitational microlensing ( arXiv:astro-ph/0305166 ). Of particular interest is the construction and the behavior of rational functions $$R(z)$$ R ( z ) that are extremal in the sense that $$R(z)-\overline{z}$$ R ( z ) - z ¯ has the maximal possible number of $$5(d-1)$$ 5 ( d - 1 ) zeros.

Journal

Computational Methods and Function TheorySpringer Journals

Published: Aug 7, 2014

References